Question about fitch 6.19 proving A or C from premises A or B and -B or C

Question

How to prove A or C from premises A or B and -B or C. Am using fitch and have been stuck on this for an hour

Answer 1

Our goal is a disjunction. Working forward (from the premises) seems a good option. As A v B and ¬B v C both have a disjunction as its main logical connective, we will attempt to use Disjunction Elimination rule. The proof skeleton, would be:

1.  | A v B
2.  |_ ¬B v C
3.  | |_ A
4.  | | A v C       v Intro: 3
5.  | |_ B
6.  | | |_ ¬B
7.  | | | ⊥        ⊥ Intro: 6,5
8.  | | | ...
9.  | | |_ C
10. | | | ...
11. | | ...
12. | A v C         v Elim: 3-4, 5-11

Can you complete the proof, now ?

Answer 2

1. (A v B)
2. -(B v C)
3.         | A___        ASSUME
4.         | (A v C)     [v-introduction, 3]
5.         | B___       ASSUME
6.            | -B___    ASSUME
7.            | ⊥           [⊥-introduction, 5,6]
8.            | (A v C)   [⊥-elimination, 7]
9.            | C___     ASSUME
10.            | (A v C)   [v-introduction, 9]
11.         | (A v C)      [v-elimination, 2, 6-8, 9-10]
12. (A v C)                [v-elimination, 1, 3-4, 5-11]